Eta-expansions in Fω

  • Neil Ghani
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1258)


The use of expansionary η-rewrite rules in typed λ-calculi has become increasingly common in recent years as their advantages over contractive η-rewrite rules have become more apparent. Not only does one obtain simultaneously the decidability of βη-equality and a natural construction of the long βη-normal forms, but rewrite relations using expansions retain key properties when combined with first order rewrite systems, generalise more easily to other type constructors and are supported by a categorical theory of reduction.

However, one area where η-contractions have until now remained the only possibility is in the more powerful type theories of the λ-cube. This paper begins to rectify this situation by considering a higher order polymorphic λ-calculus known as Fω.


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  1. 1.
    Y. Akama. On Mints' reduction for ccc-calculus. In Typed λ-Calculus and Applications, volume 664 of Lecture Notes in Computer Science, pages 1–12. Springer Verlag, 1993.Google Scholar
  2. 2.
    H. Barendregt. The Lambda Calculus: Its Syntax and Semantics (revised edition). Number 103 in Studies in Logic and the Foundations of Mathematics. North Holland, 1984.Google Scholar
  3. 3.
    V. Breazu-Tannen and J. Gallier. Polymorphic rewriting preserves algebraic strong normalisation. Theoretical Computer Science, 83:3–28, 1991.Google Scholar
  4. 4.
    V. Breazu-Tannen and J. Gallier. Polymorphic rewriting preserves algebraic confluence. Information and Computation, 114:1–29, 1994.Google Scholar
  5. 5.
    R. Di Cosmo and D. Kesner. Simulating expansions without expansions. Mathematical Structures in Computer Science, 4:1–48, 1994.Google Scholar
  6. 6.
    R. Di Cosmo and D. Kesner. Combining algebraic rewriting, extensional λ-calculi and fixpoints. In TCS, 1995.Google Scholar
  7. 7.
    D. Dougherty. Some λ-calculi with categorical sums and products. In Rewriting Techniques and Applications, volume 690 of Lecture Notes in Computer Science, pages 137–151. Springer Verlag, 1993.Google Scholar
  8. 8.
    G. Dowek. On the defintion of the η-long normal form in type systems of the cube. Informal proceedings of the 1993 Workshop on Types for Proofs and Programs, 1993.Google Scholar
  9. 9.
    J. Gallier. On girard's candidats de reductibilite. Logic and Computer Science, pages 123–203, 1990.Google Scholar
  10. 10.
    H. Geuvers. The Church-Rosser property for βη-reduction in typed λ-calculi. In LICS, pages 453–460. IEEE, 1992.Google Scholar
  11. 11.
    N. Ghani. Adjoint Rewriting. PhD thesis, University of Edinburgh, Department of Computer Science, 1995.Google Scholar
  12. 12.
    N. Ghani. βη-equality for coproducts. In Typed λ-calculus and Applications, number 902 in Lecture Notes in Computer Science, pages 171–185. Springer Verlag, 1995.Google Scholar
  13. 13.
    G. Huet. Résolution d'équations dans des langages d'ordre 1,2,⋯, ω. Thése d'Etat, Université de Paris VII, 1976.Google Scholar
  14. 14.
    C. B. Jay. Modelling reduction in confluent categories. In Applications of Categories in Computer Science, volume 177. London Mathematical Society Lecture Note Series, 1992.Google Scholar
  15. 15.
    C. B. Jay and N. Ghani. The virtues of eta-expansion. Journal of Functional Programming, Volume 5(2), April 1995, pages 135–154. CUP 1995.Google Scholar
  16. 16.
    G. E. Mints. Teorija categorii i teoria dokazatelstv.i. Aktualnye problemy logiki i metodologii nauky, pages 252–278, 1979.Google Scholar
  17. 17.
    D. Prawitz. Ideas and results in proof theory. In J.E. Fenstad, editor, Proc. 2nd Scandinavian Logic Symposium, pages 235–307. North Holland, 1971.Google Scholar
  18. 18.
    R. A. G. Seely. Modelling computations: A 2-categorical framework. In Proc. 2nd Annual Symposium on Logic in Computer Science. IEEE publications, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Neil Ghani
    • 1
  1. 1.Ecole Normale SuperieureLIENS-DMIParis Cedex 05France

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